Question

determine whether \\(\frac{3^{5m}}{3^{3m}}\\) is equivalent to each of the following expressions.
equivalent not equivalent
\\(9^m\\)
\\(3^{2m}\\)
\\(81^{0.5m}\\)

Explanation:

Step1: Simplify the original expression

Use exponent rule $\frac{a^x}{a^y}=a^{x-y}$:
$\frac{3^{5m}}{3^{3m}}=3^{5m-3m}=3^{2m}$

Step2: Compare with $9^m$

Rewrite $9$ as $3^2$:
$9^m=(3^2)^m=3^{2m}$

Step3: Compare with $3^{2m}$

Directly match the simplified original expression:
$3^{2m}=3^{2m}$

Step4: Compare with $81^{0.5m}$

Rewrite $81$ as $3^4$:
$81^{0.5m}=(3^4)^{0.5m}=3^{4\times0.5m}=3^{2m}$

Answer:

• $9^m$: Equivalent
• $3^{2m}$: Equivalent
• $81^{0.5m}$: Equivalent